\(\int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx\) [1621]

Optimal result

Integrand size = 22, antiderivative size = 263 \[ \int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx=\frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}+\frac {x^3 \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{3 c \sqrt [3]{a+b x^2}}+\frac {\sqrt {3} c^2 \arctan \left (\frac {1+\frac {2 d^{2/3} \sqrt [3]{a+b x^2}}{\sqrt [3]{b c^2+a d^2}}}{\sqrt {3}}\right )}{2 d^{7/3} \sqrt [3]{b c^2+a d^2}}-\frac {c^2 \log \left (c^2-d^2 x^2\right )}{4 d^{7/3} \sqrt [3]{b c^2+a d^2}}+\frac {3 c^2 \log \left (\sqrt [3]{b c^2+a d^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{4 d^{7/3} \sqrt [3]{b c^2+a d^2}} \] Output:

3/4*(b*x^2+a)^(2/3)/b/d+1/3*x^3*(1+b*x^2/a)^(1/3)*AppellF1(3/2,1,1/3,5/2,d 
^2*x^2/c^2,-b*x^2/a)/c/(b*x^2+a)^(1/3)+1/2*3^(1/2)*c^2*arctan(1/3*(1+2*d^( 
2/3)*(b*x^2+a)^(1/3)/(a*d^2+b*c^2)^(1/3))*3^(1/2))/d^(7/3)/(a*d^2+b*c^2)^( 
1/3)-1/4*c^2*ln(-d^2*x^2+c^2)/d^(7/3)/(a*d^2+b*c^2)^(1/3)+3/4*c^2*ln((a*d^ 
2+b*c^2)^(1/3)-d^(2/3)*(b*x^2+a)^(1/3))/d^(7/3)/(a*d^2+b*c^2)^(1/3)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx=\int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx \] Input:

Integrate[x^2/((c + d*x)*(a + b*x^2)^(1/3)),x]
 

Output:

Integrate[x^2/((c + d*x)*(a + b*x^2)^(1/3)), x]
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(844\) vs. \(2(263)=526\).

Time = 0.85 (sec) , antiderivative size = 844, normalized size of antiderivative = 3.21, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {605, 241, 605, 233, 504, 334, 333, 353, 67, 16, 833, 760, 1082, 217, 2418}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2}{\sqrt [3]{a+b x^2} (c+d x)} \, dx\)

\(\Big \downarrow \) 605

\(\displaystyle \frac {\int \frac {x}{\sqrt [3]{b x^2+a}}dx}{d}-\frac {c \int \frac {x}{(c+d x) \sqrt [3]{b x^2+a}}dx}{d}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \int \frac {x}{(c+d x) \sqrt [3]{b x^2+a}}dx}{d}\)

\(\Big \downarrow \) 605

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {\int \frac {1}{\sqrt [3]{b x^2+a}}dx}{d}-\frac {c \int \frac {1}{(c+d x) \sqrt [3]{b x^2+a}}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 b d x}-\frac {c \int \frac {1}{(c+d x) \sqrt [3]{b x^2+a}}dx}{d}\right )}{d}\)

\(\Big \downarrow \) 504

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 b d x}-\frac {c \left (c \int \frac {1}{\sqrt [3]{b x^2+a} \left (c^2-d^2 x^2\right )}dx-d \int \frac {x}{\sqrt [3]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 b d x}-\frac {c \left (\frac {c \sqrt [3]{\frac {b x^2}{a}+1} \int \frac {1}{\sqrt [3]{\frac {b x^2}{a}+1} \left (c^2-d^2 x^2\right )}dx}{\sqrt [3]{a+b x^2}}-d \int \frac {x}{\sqrt [3]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 333

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{a+b x^2}}-d \int \frac {x}{\sqrt [3]{b x^2+a} \left (c^2-d^2 x^2\right )}dx\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{a+b x^2}}-\frac {1}{2} d \int \frac {1}{\sqrt [3]{b x^2+a} \left (c^2-d^2 x^2\right )}dx^2\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{a+b x^2}}-\frac {1}{2} d \left (\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c^2+a d^2}}{d^{2/3}}-\sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \int \frac {1}{x^4+\frac {\left (b c^2+a d^2\right )^{2/3}}{d^{4/3}}+\frac {\sqrt [3]{b c^2+a d^2} \sqrt [3]{b x^2+a}}{d^{2/3}}}d\sqrt [3]{b x^2+a}}{2 d^2}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{a+b x^2}}-\frac {1}{2} d \left (-\frac {3 \int \frac {1}{x^4+\frac {\left (b c^2+a d^2\right )^{2/3}}{d^{4/3}}+\frac {\sqrt [3]{b c^2+a d^2} \sqrt [3]{b x^2+a}}{d^{2/3}}}d\sqrt [3]{b x^2+a}}{2 d^2}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \log \left (\sqrt [3]{a d^2+b c^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{a+b x^2}}-\frac {1}{2} d \left (-\frac {3 \int \frac {1}{x^4+\frac {\left (b c^2+a d^2\right )^{2/3}}{d^{4/3}}+\frac {\sqrt [3]{b c^2+a d^2} \sqrt [3]{b x^2+a}}{d^{2/3}}}d\sqrt [3]{b x^2+a}}{2 d^2}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \log \left (\sqrt [3]{a d^2+b c^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{a+b x^2}}-\frac {1}{2} d \left (-\frac {3 \int \frac {1}{x^4+\frac {\left (b c^2+a d^2\right )^{2/3}}{d^{4/3}}+\frac {\sqrt [3]{b c^2+a d^2} \sqrt [3]{b x^2+a}}{d^{2/3}}}d\sqrt [3]{b x^2+a}}{2 d^2}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \log \left (\sqrt [3]{a d^2+b c^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{a+b x^2}}-\frac {1}{2} d \left (\frac {3 \int \frac {1}{-x^4-3}d\left (\frac {2 \sqrt [3]{b x^2+a} d^{2/3}}{\sqrt [3]{b c^2+a d^2}}+1\right )}{d^{4/3} \sqrt [3]{a d^2+b c^2}}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \log \left (\sqrt [3]{a d^2+b c^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 \left (a+b x^2\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{a+b x^2}}-\frac {1}{2} d \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 d^{2/3} \sqrt [3]{a+b x^2}}{\sqrt [3]{a d^2+b c^2}}+1}{\sqrt {3}}\right )}{d^{4/3} \sqrt [3]{a d^2+b c^2}}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}-\frac {3 \log \left (\sqrt [3]{a d^2+b c^2}-d^{2/3} \sqrt [3]{a+b x^2}\right )}{2 d^{4/3} \sqrt [3]{a d^2+b c^2}}\right )\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 \left (b x^2+a\right )^{2/3}}{4 b d}-\frac {c \left (\frac {3 \sqrt {b x^2} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b x^2+a}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{b x^2+a} \sqrt [3]{a}+\left (b x^2+a\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b x^2+a}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b x^2+a}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{b x^2+a} \sqrt [3]{a}+\left (b x^2+a\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b x^2+a}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )}{2 b d x}-\frac {c \left (\frac {x \sqrt [3]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{c \sqrt [3]{b x^2+a}}-\frac {1}{2} d \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b x^2+a} d^{2/3}}{\sqrt [3]{b c^2+a d^2}}+1}{\sqrt {3}}\right )}{d^{4/3} \sqrt [3]{b c^2+a d^2}}+\frac {\log \left (c^2-d^2 x^2\right )}{2 d^{4/3} \sqrt [3]{b c^2+a d^2}}-\frac {3 \log \left (\sqrt [3]{b c^2+a d^2}-d^{2/3} \sqrt [3]{b x^2+a}\right )}{2 d^{4/3} \sqrt [3]{b c^2+a d^2}}\right )\right )}{d}\right )}{d}\)

Input:

Int[x^2/((c + d*x)*(a + b*x^2)^(1/3)),x]
 

Output:

(3*(a + b*x^2)^(2/3))/(4*b*d) - (c*((3*Sqrt[b*x^2]*((-2*Sqrt[b*x^2])/((1 - 
 Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3 
)*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) 
+ (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*Ellipt 
icE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1 
/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[b*x^2]*Sqrt[-((a^(1/3)*( 
a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^ 
2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*a^(1/3)*(a^(1/3) - (a + b*x^2)^( 
1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - 
 Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])* 
a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], 
 -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[b*x^2]*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x 
^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])))/(2*b*d*x) - 
 (c*((x*(1 + (b*x^2)/a)^(1/3)*AppellF1[1/2, 1/3, 1, 3/2, -((b*x^2)/a), (d^ 
2*x^2)/c^2])/(c*(a + b*x^2)^(1/3)) - (d*(-((Sqrt[3]*ArcTan[(1 + (2*d^(2/3) 
*(a + b*x^2)^(1/3))/(b*c^2 + a*d^2)^(1/3))/Sqrt[3]])/(d^(4/3)*(b*c^2 + a*d 
^2)^(1/3))) + Log[c^2 - d^2*x^2]/(2*d^(4/3)*(b*c^2 + a*d^2)^(1/3)) - (3*Lo 
g[(b*c^2 + a*d^2)^(1/3) - d^(2/3)*(a + b*x^2)^(1/3)])/(2*d^(4/3)*(b*c^2 + 
a*d^2)^(1/3))))/2))/d))/d
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[ 
(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p, q}, x] && 
NeQ[b*c - a*d, 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

rule 605

Int[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] 
:> Simp[1/d   Int[x^(m - 1)*(a + b*x^2)^p, x], x] - Simp[c/d   Int[x^(m - 1 
)*((a + b*x^2)^p/(c + d*x)), x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 
 0] && LtQ[-1, p, 0]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {x^{2}}{\left (d x +c \right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}}d x\]

Input:

int(x^2/(d*x+c)/(b*x^2+a)^(1/3),x)
 

Output:

int(x^2/(d*x+c)/(b*x^2+a)^(1/3),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate(x^2/(d*x+c)/(b*x^2+a)^(1/3),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx=\int \frac {x^{2}}{\sqrt [3]{a + b x^{2}} \left (c + d x\right )}\, dx \] Input:

integrate(x**2/(d*x+c)/(b*x**2+a)**(1/3),x)
 

Output:

Integral(x**2/((a + b*x**2)**(1/3)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate(x^2/(d*x+c)/(b*x^2+a)^(1/3),x, algorithm="maxima")
 

Output:

integrate(x^2/((b*x^2 + a)^(1/3)*(d*x + c)), x)
 

Giac [F]

\[ \int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate(x^2/(d*x+c)/(b*x^2+a)^(1/3),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate(x^2/((b*x^2 + a)^(1/3)*(d*x + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx=\int \frac {x^2}{{\left (b\,x^2+a\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \] Input:

int(x^2/((a + b*x^2)^(1/3)*(c + d*x)),x)
 

Output:

int(x^2/((a + b*x^2)^(1/3)*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {x^2}{(c+d x) \sqrt [3]{a+b x^2}} \, dx=\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} c +\left (b \,x^{2}+a \right )^{\frac {1}{3}} d x}d x \] Input:

int(x^2/(d*x+c)/(b*x^2+a)^(1/3),x)
 

Output:

int(x**2/((a + b*x**2)**(1/3)*c + (a + b*x**2)**(1/3)*d*x),x)